Why can't I export my rigs?
From the wikipedia wavefront obj page -


  The OBJ file format is a simple data-format that represents 3D geometry alone — namely, the position of each vertex, the UV position of each texture coordinate vertex, vertex normals, and the faces that make each polygon defined as a list of vertices, and texture vertices


If you look in the unreal docs under Managing content they only mention fbx as a format for importing static or skeletal meshes. Blender includes an fbx import/export addon, if you don't see the fbx export option then you will need to enable the addon.

There is also an addon available that eases the blender -> UE process, available on github.